Finite Group Databasebeta

← C3×#10605015 #14297189

Subgroups — C3×#10605015 #14297189

Order 1920.

1240 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D5×H(5,𝔽2)C211yesmaximal, characteristic
2C3×#65793C212yesmaximal, complemented
3C3×D5⋊#104C212yesmaximal, complemented
4C3×#65780C211yesmaximal, characteristic, complemented
5C3×D5⋊#100C211yesmaximal, characteristic, complemented
6#10605015C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×H(5,𝔽2)C411yescharacteristic, complemented
8C3×#4149—22no—
9C3×#4149C412yescomplemented
10C3×D5×#41—22no—
11C3×D5×#41C2212yes—
12C3×#4144—22no—
13C3×#4144—22no—
14C3×#4144C411yescharacteristic, complemented
15C2×C3×D4×D5—22no—
16C2×C3×D4×D5—22no—
17C2×C3×D4×D5C2211yescharacteristic
18C2×C3×C22⋊AGL(1,𝔽5)—22no—
19C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic, complemented
20C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic, complemented
21C3×D5⋊M16C2212yescomplemented
22C3×#4013—22no—
23C3×#4010—22no—
24C3×#4009—22no—
25C3×#4008—22no—
26C3×#4007—22no—
27C3×#1792—51nomaximal, Hall
28D5×H(5,𝔽2)C2×C311yescharacteristic
29#65793C2×C312yescomplemented
30D5⋊#104C2×C312yescomplemented
31#65780C2×C311yescharacteristic, complemented
32D5⋊#100C2×C311yescharacteristic, complemented
33C23×C3×D5—22no—
34C23×C3×D5D412yes—
35C22×C3×AGL(1,𝔽5)—21no—
36C22×C3×AGL(1,𝔽5)—21no—
37C3×C5×#41—22no—
38C3×C5×#41C2×C412yes—
39C2×C3×C5×D4—22no—
40C2×C3×C5×D4—22no—
41C2×C3×C5×D4C2×C411yescharacteristic
42C2×C3×C22⋊D5—22no—
43C2×C3×C22⋊D5—24no—
44C2×C3×C22⋊D5—22no—
45C2×C3×C22⋊D5D412yes—
46C3×#713—22no—
47C3×#713D412yes—
48C3×Q8×D5D412yes—
49C3×#711—24no—
50C3×#711—22no—
51C3×#711—22no—
52C3×#711D412yes—
53C3×D4×D5—24no—
54C3×D4×D5—24no—
55C3×D4×D5—24no—
56C3×D4×D5—24no—
57C3×D4×D5—22no—
58C3×D4×D5—22no—
59C3×D4×D5—22no—
60C3×D4×D5—22no—
61C3×D4×D5—22no—
62C3×D4×D5D412yes—
63C3×#709—24no—
64C3×#709—22no—
65C3×#709—22no—
66C3×#709C2×C412yes—
67C2×C3×D20—22no—
68C2×C3×D20—22no—
69C2×C3×D20C2×C411yescharacteristic
70C2×C3×C4×D5—22no—
71C2×C3×C4×D5—22no—
72C2×C3×C4×D5C2311yescharacteristic
73C3×C22⋊AGL(1,𝔽5)—44no—
74C3×#704—22no—
75C3×C4⋊AGL(1,𝔽5)—22no—
76C3×C4×AGL(1,𝔽5)—22no—
77C3×#700—22no—
78C3×D5⋊C8—22no—
79C2×C3×H(5,𝔽2)—51no—
80C2×C3×#104—52no—
81C3×#417—52no—
82C2×C3×#100—51no—
83C3×#390—51no—
84C5×H(5,𝔽2)C3×C411yescharacteristic, complemented
85#4149—22no—
86#4149C3×C412yescomplemented
87D5×#41—22no—
88D5×#41C22×C312yes—
89#4144—22no—
90#4144—22no—
91#4144C3×C411yescharacteristic, complemented
92C2×D4×D5—22no—
93C2×D4×D5—22no—
94C2×D4×D5C22×C311yescharacteristic
95C2×C22⋊AGL(1,𝔽5)—22no—
96C2×C4⋊AGL(1,𝔽5)C22×C311yescharacteristic, complemented
97C2×C4×AGL(1,𝔽5)C22×C311yescharacteristic, complemented
98D5⋊M16C22×C312yescomplemented
99#4013—22no—
100#4010—22no—
101#4009—22no—
102#4008—22no—
103#4007—22no—
104#1792—51noSylow
105C23×C3×C5—22no—
106C23×C3×C5C22⋊C412yes—
107C22×C3×D5—24no—
108C22×C3×D5—22no—
109C22×C3×D5—22no—
110C22×C3×D5—22no—
111C22×C3×D5—22no—
112C22×C3×D5—42no—
113C22×C3×D5—42no—
114C22×C3×D5—82no—
115C22×C3×D5—82no—
116C22×C3×D5C22⋊C412yes—
117C22×C3×D5C2×D411yescharacteristic
118C2×C3×AGL(1,𝔽5)—22no—
119C2×C3×AGL(1,𝔽5)—41no—
120C2×C3×AGL(1,𝔽5)—41no—
121C2×C3×AGL(1,𝔽5)—42no—
122C2×C3×AGL(1,𝔽5)—41no—
123C3×C5×Q8C22⋊C412yescomplemented
124C3×C5×D4—24no—
125C3×C5×D4—22no—
126C3×C5×D4—22no—
127C3×C5×D4C22⋊C412yescomplemented
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5C22×C411yescharacteristic
131C3×C22⋊D5—24no—
132C3×C22⋊D5—24no—
133C3×C22⋊D5—22no—
134C3×C22⋊D5—22no—
135C3×C22⋊D5—24no—
136C3×C22⋊D5—22no—
137C2×C3×Dic5—22no—
138C2×C3×Dic5—22no—
139C2×C3×Dic5C2×D411yescharacteristic
140C3×D20—24no—
141C3×D20—22no—
142C3×D20—22no—
143C3×D20C22⋊C412yescomplemented
144C3×C4×D5—24no—
145C3×C4×D5—22no—
146C3×C4×D5—22no—
147C3×C4×D5—22no—
148C3×C4×D5C2×D412yes—
149C3×C4×D5C2×D412yes—
150C3×Dic10—22no—
151C3×Dic10C22⋊C412yescomplemented
152C3×#170—42no—
153C3×H(5,𝔽2)—52no—
154C3×H(5,𝔽2)—51no—
155C3×H(5,𝔽2)—102no—
156C3×H(5,𝔽2)—102no—
157C3×H(5,𝔽2)—102no—
158C3×H(5,𝔽2)AGL(1,𝔽5)11yescharacteristic, complemented
159C2×C3×#41—52no—
160C2×C3×#41—102no—
161C22×C3×D4—51no—
162C22×C3×D4—102no—
163C22×C3×D4—102no—
164C2×C3×M16—52no—
165C2×C3×C4⋊C4—51no—
166C2×C3×C22⋊C4—102no—
167C2×C3×C42—51no—
168C3×#104—102no—
169C3×#104—102no—
170C3×#103—102no—
171C3×#102—102no—
172C3×#100—102no—
173C23×D5—22no—
174C23×D5C3×D412yes—
175C22×AGL(1,𝔽5)—21no—
176C22×AGL(1,𝔽5)—21no—
177C5×#41—22no—
178C5×#41C2×C3×C412yes—
179C2×C5×D4—22no—
180C2×C5×D4—22no—
181C2×C5×D4C2×C3×C411yescharacteristic
182C2×C22⋊D5—22no—
183C2×C22⋊D5—24no—
184C2×C22⋊D5—22no—
185C2×C22⋊D5C3×D412yes—
186#713—22no—
187#713C3×D412yes—
188Q8×D5C3×D412yes—
189#711—24no—
190#711—22no—
191#711—22no—
192#711C3×D412yes—
193D4×D5—24no—
194D4×D5—24no—
195D4×D5—24no—
196D4×D5—24no—
197D4×D5—22no—
198D4×D5—22no—
199D4×D5—22no—
200D4×D5—22no—
201D4×D5—22no—
202D4×D5C3×D412yes—
203#709—24no—
204#709—22no—
205#709—22no—
206#709C2×C3×C412yes—
207C2×D20—22no—
208C2×D20—22no—
209C2×D20C2×C3×C411yescharacteristic
210C2×C4×D5—22no—
211C2×C4×D5—22no—
212C2×C4×D5C23×C311yescharacteristic
213C22⋊AGL(1,𝔽5)—44no—
214#704—22no—
215C4⋊AGL(1,𝔽5)—22no—
216C4×AGL(1,𝔽5)—22no—
217#700—22no—
218D5⋊C8—22no—
219C2×H(5,𝔽2)—51no—
220C2×#104—52no—
221#417—52no—
222C2×#100—51no—
223#390—51no—
224C22×C3×C5—22no—
225C22×C3×C5—22no—
226C22×C3×C5—42no—
227C22×C3×C5—82no—
228C22×C3×C5C2×C22⋊C411yescharacteristic
229C2×C3×D5—21no—
230C2×C3×D5—21no—
231C2×C3×D5—22no—
232C2×C3×D5—22no—
233C2×C3×D5—42no—
234C2×C3×D5—42no—
235C2×C3×D5—42no—
236C2×C3×D5—42no—
237C2×C3×D5—42no—
238C2×C3×D5—82no—
239C2×C3×D5—84no—
240C2×C3×D5—82no—
241C2×C3×D5#12011yescharacteristic
242C2×C3×D5#12011yescharacteristic
243C3×AGL(1,𝔽5)—42no—
244C3×AGL(1,𝔽5)—81no—
245C3×C4×C5—22no—
246C3×C4×C5C2×C22⋊C412yes—
247C3×Dic5—22no—
248C3×Dic5#12012yes—
249C24×C3—52no—
250C24×C3—102no—
251C3×#41—22no—
252C3×#41—52no—
253C3×#41—52no—
254C3×#41—52no—
255C3×#41—102no—
256C3×#41—102no—
257C3×#41—104no—
258C3×#41—102no—
259C3×#41—102no—
260C3×#41—104no—
261C3×#41—102no—
262C3×#41C2×AGL(1,𝔽5)12yes—
263C2×C3×Q8—52no—
264C2×C3×D4—22no—
265C2×C3×D4—22no—
266C2×C3×D4—52no—
267C2×C3×D4—52no—
268C2×C3×D4—51no—
269C2×C3×D4—102no—
270C2×C3×D4—102no—
271C2×C3×D4—102no—
272C2×C3×D4—102no—
273C2×C3×D4—102no—
274C2×C3×D4—102no—
275C2×C3×D4—104no—
276C2×C3×D4—104no—
277C2×C3×D4—104no—
278C2×C3×D4—104no—
279C2×C3×D4—104no—
280C2×C3×D4—102no—
281C2×C3×D4—102no—
282C2×C3×D4—102no—
283C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
284C22×C3×C4—51no—
285C22×C3×C4—102no—
286C22×C3×C4—102no—
287C22×C3×C4—101no—
288C22×C3×C4—101no—
289C3×M16—102no—
290C3×M16—102no—
291C2×C3×C8—102no—
292C3×C4⋊C4—102no—
293C3×C22⋊C4—204no—
294C3×C42—102no—
295C23×C5—22no—
296C23×C5C3×C22⋊C412yes—
297C22×D5—24no—
298C22×D5—22no—
299C22×D5—22no—
300C22×D5—22no—
301C22×D5—22no—
302C22×D5—42no—
303C22×D5—42no—
304C22×D5—82no—
305C22×D5—82no—
306C22×D5C3×C22⋊C412yes—
307C22×D5C2×C3×D411yescharacteristic
308C2×AGL(1,𝔽5)—22no—
309C2×AGL(1,𝔽5)—41no—
310C2×AGL(1,𝔽5)—41no—
311C2×AGL(1,𝔽5)—42no—
312C2×AGL(1,𝔽5)—41no—
313C5×Q8C3×C22⋊C412yescomplemented
314C5×D4—24no—
315C5×D4—22no—
316C5×D4—22no—
317C5×D4C3×C22⋊C412yescomplemented
318C2×C4×C5—22no—
319C2×C4×C5—22no—
320C2×C4×C5C22×C3×C411yescharacteristic
321C22⋊D5—24no—
322C22⋊D5—24no—
323C22⋊D5—22no—
324C22⋊D5—22no—
325C22⋊D5—24no—
326C22⋊D5—22no—
327C2×Dic5—22no—
328C2×Dic5—22no—
329C2×Dic5C2×C3×D411yescharacteristic
330D20—24no—
331D20—22no—
332D20—22no—
333D20C3×C22⋊C412yescomplemented
334C4×D5—24no—
335C4×D5—22no—
336C4×D5—22no—
337C4×D5—22no—
338C4×D5C2×C3×D412yes—
339C4×D5C2×C3×D412yes—
340Dic10—22no—
341Dic10C3×C22⋊C412yescomplemented
342#170—42no—
343H(5,𝔽2)—52no—
344H(5,𝔽2)—51no—
345H(5,𝔽2)—102no—
346H(5,𝔽2)—102no—
347H(5,𝔽2)—102no—
348H(5,𝔽2)C3×AGL(1,𝔽5)11yescharacteristic, complemented
349C2×#41—52no—
350C2×#41—102no—
351C22×D4—51no—
352C22×D4—102no—
353C22×D4—102no—
354C2×M16—52no—
355C2×C4⋊C4—51no—
356C2×C22⋊C4—102no—
357C2×C42—51no—
358#104—102no—
359#104—102no—
360#103—102no—
361#102—102no—
362#100—102no—
363C2×C3×C5—21no—
364C2×C3×C5—42no—
365C2×C3×C5—42no—
366C2×C3×C5#37911yescharacteristic
367C3×D5—21no—
368C3×D5—42no—
369C3×D5—42no—
370C3×D5#45311yescharacteristic
371C3×D5#45511yescharacteristic
372C23×C3—22no—
373C23×C3—51no—
374C23×C3—52no—
375C23×C3—102no—
376C23×C3—102no—
377C23×C3—102no—
378C23×C3—102no—
379C23×C3—104no—
380C23×C3—202no—
381C23×C3—202no—
382C23×C3—402no—
383C23×C3—402no—
384C23×C3C22⋊AGL(1,𝔽5)12yes—
385C3×Q8—52no—
386C3×Q8—102no—
387C3×Q8C22⋊AGL(1,𝔽5)12yescomplemented
388C3×D4—24no—
389C3×D4—22no—
390C3×D4—22no—
391C3×D4—52no—
392C3×D4—102no—
393C3×D4—104no—
394C3×D4—104no—
395C3×D4—102no—
396C3×D4—102no—
397C3×D4—104no—
398C3×D4—104no—
399C3×D4—102no—
400C3×D4—102no—
401C3×D4C22⋊AGL(1,𝔽5)12yescomplemented
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—51no—
405C2×C3×C4—52no—
406C2×C3×C4—52no—
407C2×C3×C4—102no—
408C2×C3×C4—102no—
409C2×C3×C4—104no—
410C2×C3×C4—102no—
411C2×C3×C4—102no—
412C2×C3×C4—102no—
413C2×C3×C4—102no—
414C2×C3×C4—201no—
415C2×C3×C4—201no—
416C2×C3×C4—201no—
417C2×C3×C4—202no—
418C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
419C3×C8—202no—
420C22×C5—22no—
421C22×C5—22no—
422C22×C5—42no—
423C22×C5—82no—
424C22×C5C2×C3×C22⋊C411yescharacteristic
425C2×D5—21no—
426C2×D5—21no—
427C2×D5—22no—
428C2×D5—22no—
429C2×D5—42no—
430C2×D5—42no—
431C2×D5—42no—
432C2×D5—42no—
433C2×D5—42no—
434C2×D5—82no—
435C2×D5—84no—
436C2×D5—82no—
437C2×D5C3×#12011yescharacteristic
438C2×D5C3×#12011yescharacteristic
439AGL(1,𝔽5)—42no—
440AGL(1,𝔽5)—81no—
441C4×C5—22no—
442C4×C5C2×C3×C22⋊C412yes—
443Dic5—22no—
444Dic5C3×#12012yes—
445C24—52no—
446C24—102no—
447#41—22no—
448#41—52no—
449#41—52no—
450#41—52no—
451#41—102no—
452#41—102no—
453#41—104no—
454#41—102no—
455#41—102no—
456#41—104no—
457#41—102no—
458#41C2×C3×AGL(1,𝔽5)12yes—
459C2×Q8—52no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—52no—
463C2×D4—52no—
464C2×D4—51no—
465C2×D4—102no—
466C2×D4—102no—
467C2×D4—102no—
468C2×D4—102no—
469C2×D4—102no—
470C2×D4—102no—
471C2×D4—104no—
472C2×D4—104no—
473C2×D4—104no—
474C2×D4—104no—
475C2×D4—104no—
476C2×D4—102no—
477C2×D4—102no—
478C2×D4—102no—
479C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
480C22×C4—51no—
481C22×C4—102no—
482C22×C4—102no—
483C22×C4—101no—
484C22×C4—101no—
485M16—102no—
486M16—102no—
487C2×C8—102no—
488C4⋊C4—102no—
489C22⋊C4—204no—
490C42—102no—
491C3×C5#179211yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—42no—
495C22×C3—51no—
496C22×C3—51no—
497C22×C3—82no—
498C22×C3—101no—
499C22×C3—101no—
500C22×C3—102no—
501C22×C3—102no—
502C22×C3—202no—
503C22×C3—202no—
504C22×C3—202no—
505C22×C3—202no—
506C22×C3—202no—
507C22×C3—402no—
508C22×C3—402no—
509C22×C3—404no—
510C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
511C3×C4—22no—
512C3×C4—52no—
513C3×C4—102no—
514C3×C4—202no—
515C3×C4—401no—
516C3×C4C2×C22⋊AGL(1,𝔽5)12yes—
517C2×C5—21no—
518C2×C5—42no—
519C2×C5—42no—
520C2×C5C3×#37911yescharacteristic
521D5—21no—
522D5—42no—
523D5—42no—
524D5C3×#45311yescharacteristic
525D5C3×#45511yescharacteristic
526C23—22no—
527C23—51no—
528C23—52no—
529C23—102no—
530C23—102no—
531C23—102no—
532C23—102no—
533C23—104no—
534C23—202no—
535C23—202no—
536C23—402no—
537C23—402no—
538C23C3×C22⋊AGL(1,𝔽5)12yes—
539Q8—52no—
540Q8—102no—
541Q8C3×C22⋊AGL(1,𝔽5)12yescomplemented
542D4—24no—
543D4—22no—
544D4—22no—
545D4—52no—
546D4—102no—
547D4—104no—
548D4—104no—
549D4—102no—
550D4—102no—
551D4—104no—
552D4—104no—
553D4—102no—
554D4—102no—
555D4C3×C22⋊AGL(1,𝔽5)12yescomplemented
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—51no—
559C2×C4—52no—
560C2×C4—52no—
561C2×C4—102no—
562C2×C4—102no—
563C2×C4—104no—
564C2×C4—102no—
565C2×C4—102no—
566C2×C4—102no—
567C2×C4—102no—
568C2×C4—201no—
569C2×C4—201no—
570C2×C4—201no—
571C2×C4—202no—
572C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
573C8—202no—
574C2×C3—21no—
575C2×C3—42no—
576C2×C3—42no—
577C2×C3—51no—
578C2×C3—51no—
579C2×C3—101no—
580C2×C3—202no—
581C2×C3—202no—
582C2×C3#6580011yescharacteristic
583C5C3×#179211yescharacteristic, complemented, Sylow
584C22—22no—
585C22—22no—
586C22—42no—
587C22—51no—
588C22—51no—
589C22—82no—
590C22—101no—
591C22—101no—
592C22—102no—
593C22—102no—
594C22—202no—
595C22—202no—
596C22—202no—
597C22—202no—
598C22—202no—
599C22—402no—
600C22—402no—
601C22—404no—
602C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
603C4—22no—
604C4—52no—
605C4—102no—
606C4—202no—
607C4—401no—
608C4C2×C3×C22⋊AGL(1,𝔽5)12yes—
609C3#1060501511yescharacteristic, direct product factor, Sylow
610C2—21no—
611C2—42no—
612C2—42no—
613C2—51no—
614C2—51no—
615C2—101no—
616C2—202no—
617C2—202no—
618C2C3×#6580011yescharacteristic
subgroupG/H
centerC2×C3#65800
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×H(5,𝔽2)C4
socleC2×C3×C5#379
prime radicalC3×D5×H(5,𝔽2)C2
cosocleC2×C4×D5C23×C3
hypercenterC3×H(5,𝔽2)AGL(1,𝔽5)
hypocenterC5C3×#1792
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×H(5,𝔽2)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#65793C2semidirect
3. C3×D5⋊#104C2semidirect
4. C3×#65780C2semidirect
5. C3×D5⋊#100C2semidirect
6. #10605015C3direct (N × H)
7. C3×C5×H(5,𝔽2)C4semidirect
9. C3×#4149C4semidirect
14. C3×#4144C4semidirect
19. C2×C3×C4⋊AGL(1,𝔽5)C22semidirect
20. C2×C3×C4×AGL(1,𝔽5)C22semidirect
21. C3×D5⋊M16C22semidirect
29. #65793C2×C3semidirect
30. D5⋊#104C2×C3semidirect
31. #65780C2×C3semidirect
32. D5⋊#100C2×C3semidirect
84. C5×H(5,𝔽2)C3×C4semidirect
86. #4149C3×C4semidirect
91. #4144C3×C4semidirect
96. C2×C4⋊AGL(1,𝔽5)C22×C3semidirect
97. C2×C4×AGL(1,𝔽5)C22×C3semidirect
98. D5⋊M16C22×C3semidirect
123. C3×C5×Q8C22⋊C4semidirect
127. C3×C5×D4C22⋊C4semidirect
143. C3×D20C22⋊C4semidirect
151. C3×Dic10C22⋊C4semidirect
158. C3×H(5,𝔽2)AGL(1,𝔽5)semidirect
313. C5×Q8C3×C22⋊C4semidirect
317. C5×D4C3×C22⋊C4semidirect
333. D20C3×C22⋊C4semidirect
341. Dic10C3×C22⋊C4semidirect
348. H(5,𝔽2)C3×AGL(1,𝔽5)semidirect
387. C3×Q8C22⋊AGL(1,𝔽5)semidirect
401. C3×D4C22⋊AGL(1,𝔽5)semidirect
491. C3×C5#1792semidirect
541. Q8C3×C22⋊AGL(1,𝔽5)semidirect
555. D4C3×C22⋊AGL(1,𝔽5)semidirect
583. C5C3×#1792semidirect
609. C3#10605015direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×D5×H(5,𝔽2)2
C3×#657932
C3×D5⋊#1042
C3×#657802
C3×D5⋊#1002
#106050153
C3×#17925